- ; -X-661-71-443PREPRINT

THE-EQUILIBRIUM AND STABILITYOF THE GASEOUS COMPONENT

OF THE GALAXY. II.

SANFORD A. KELLMAN

NOVEMBER 1971..

1,- GODDARD SPACE FLIGHT CENTERGREENBELT, MARYLAND

12824 (NASA-TH-X-65759) THE EQUILIBRIUM ANDSTABILITY OF THE GASEOUS COMPONENT OF THEGALAXY, 2 S.A. Kellman (NASA) )Nov. 1971

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THE EQUILIBRIUM AND STABILITY OF THE GASEOUSCOMPONENT OF THE GALAXY. II.

Sanford A. KellmanDepartment of Physics and Astronomy

University of Maryland, College Park, Maryland, 20742and

NASA-Goddard Space Flight Center, Greenbelt, Maryland, 20771

ABSTRACT

A time-independent, linear, plane and axially-symmetric stability

analysis is performed on a self-gravitating,plane-parallel, isothermal

layer of non-magnetic, non-rotating gas. The gas layer is immersed

in a plane-stratified isothermal layer of stars which supply a self-

consistent gravitational field. Only the gaseous component is perturbed.

Expressions are derived for the perturbed gas potential and perturbed

gas density that satisfy both the Poisson and hydrostatic equilibrium

equations. The equation governing the size of the perturbations in the

mid-plane is found to be analogous to the one-dimensional time-independent

Schrodinger equation for a particle bound by a potential well, and

with similar boundary conditions. The radius of the neutral state is

computed numerically and compared with the Jeans' and Ledoux radius.

The inclusion of a rigid stellar component increases the Ledoux radius,

though only slightly. Isodensity contours of the neutral or

marginally unstable state are constructed. Large flattened objects

with masses of 5x106 Mb and radii of 1-2 kpc result. These numbers

are not inconsistent with the large-scale structure observed in the

gaseous component of spiral arms in the Galaxy. The possibility is

discussed that the gravitational instability of the gaseous component

excites density waves of the type described by Lin.

p~TC l~NG PAGE BLATK BOT FU, ,

3

I. INTRODUCTION

The best-known perturbation analysis with applications in galactic

astronomy was carried out by Jeans (1928) and appropriately is called

the Jeans' instability problem. The initial equilibrium state was

taken to be a self-gravitating infinite uniform gas with a uniform

gravitational potential field throughout. The last two assumptions

are, however, inconsistent with the Poisson equation. It is this

inconsistency which has prompted others to reinvestigate Jeans' analysis

with self-consistent equilibrium density and potential distributions.

Jeans applied perturbations to the momentum, continuity,

Poisson, and heat equation, and after linearization and Fourier analysis

1/2found the condition for marginal stability to be AJ = cs(r/Gpo) /,

where cs

is the isothermal sound speed of the medium and p0 is its

density. Any disturbance with length greater than AJ is gravitationally

unstable and will collapse in a finite time; any disturbance with length

smaller than AJ is stable.

The first modification to Jeans' analysis of importance to galactic

astronomy was undertaken by Ledoux (1951), who considered the more

realistic initial equilibrium state of a plane-parallel, non-rotating,

isothermal, self-gravitating gas layer (no stars). The gas density

at the plane of symmetry assumes its maximum value p0 and decreases

with distance from the mid-plane according to the well-known formula

p(z)/po = sech2(z/H), where H is the scale height of the gas layer

in the z direction. Using these self-consistent initial conditions,

Ledoux finds the condition for marginal stability to be AL = cs(27/Gpo)/2

4

precisely the same relation found by Jeans except that po is replaced

by po/2.

Goldreich and Lynden-Bell (1965) considered the gravitational

stability of a plane-stratified, self-gravitating, uniformly-rotating

disk of gas (no stars). They found that (i) pressure effects stabilize

short wavelength disturbances while rotation stabilizes long wavelength

disturbances and (ii) when the quantity ¶TGp/I402 is greater than about

1.0, where T is the mean gas density and Q is the angular velicity,

disturbances of dimension several times the layer thickness become

unstable.

In what follows, we consider a stability analysis similar to that

of Ledoux (1951), the principal difference being that the gas layer is

immersed in a rigid non-perturbable star layer which supplies a self-

consistent gravitational field. The effect of a magnetic field on the

stability will be considered in Paper III of this series (Kellman 1972a);

the effect of a magnetic field plus cosmic-ray gas will be investigated

in Paper IV (Kellman 1972b). We will not address ourselves directly

to the grand overall design of spiral structure observed in galaxies,

and discussed by Lin and Shu (1964, 1966) and Lin, Yuan, and Shu (1969).

This is probably a collective stellar phenomenon and therefore lies

outside the range of our analysis. Rather, we limit our attention

to the existence of the large-scale structure within the gaseous

component of spiral arms.

A 21-cm southern survey by McGee and Milton (1964) revealed that

the principal elements of the gaseous component of spiral arms in the

5

Galaxy are enormous flattened objects, 1-2 kpc in size and 107 M. in mass.

In addition, these structures were observed to be strung out along the

length of a spiral arm like beads on a string, their size increasing

with increasing distance from the galactic center beyond the Sun's

distance Ro. Because the projected central density of these structures

is observed to be only slightly larger than the projected ensity of

regions between the structures, a linearized stability analysis of the

type carried out below may shed light on their existence.

II. STABILITY ANALYSIS

With these ideas in mind, we proceed to examine the stability of

a self-gravitating plane-stratified distribution of gas to linear,

time-independent, plane and axially-symmetric perturbations. The plane

of symmetry is the mid-plane z=0 where the gas and star densities attain

their maximum values. The axis of symmetry is perpendicular to the

symmetry plane and arbitrarily positioned. The gas is immersed in the

gravitational field of a plane-parallel distribution of stars. Both

components of the two-fluid mixture are assumed to be isothermal, but

each has a different 'temperature'. We should stress that only the

gaseous component is perturbed; the stars merely supply a self-consistent

gravitational field which adds to the gravitational field of the gas.

This initial equilibrium state is essentially that calculated by

Kellman (1972c) in Paper I of this series. By restricting the analysis

to be time-independent, only the marginally unstable or neutral state

is determined. The inclusion of a stellar component should lend

the analysis an increased sense of reality as concerns the stability

of the gaseous component of the Galaxy.

6

The time-independent perturbations may be written as follows:

pg(r,z) = Peg(z) + Apg(r,z) (1)

pg(r,z) = Peg(Z) + Apg(rz) = <vz> pg(r,z) (2)

cPg(r,z) + cp(r,z) = Peg(Z) + pe*(Z) + AcPg(r,z). (3)

p, P, and cp are the density, pressure, and gravitational potential.

<v2z> is the mean square z turbulent gas velocity dispersion. The sub-

scripts g and * refer to the gas and star components, respectively. The

subscript e denotes the equilibrium state described above. A denotes the

perturbed quantities, functions of both r (the distance from the symmetry

axis) and z (the distance from the symmetry plane). Since only the

gaseous component is perturbed, we can write that

p*(r,z) = Pe*(Z), (4)

so that equation (3) becomes

CPg(r,z) = Peg (z) + AcPg(r,z). (5)

Equations (1), (4), and (5) are substituted into the gas hydrostatic

equilibrium equation

2<Vtz>pg Vpg = V(cg + C,), (6)

and retaining terms only to first order in Apg and AgP we find that

1 (A)-VPeg _ 1e V(APg) -_ Pg = >V(Ag). (7)Peg g PPeg

Goon (1966) has suggested the usefulness of defining a new

variable e(r,z):

e(r,z) = Apg(z) (8)Peg(z)

7

Equation (7) can now be written in terms of e:

Ve = --1 (Acg) (9)tz>

Equation (9) is integrated to give

1e = <tz> g + C, (10)

tz

where C is a constant. We apply the Laplacian operator V2 to equation (10)

with the result that

V2E = 2 V2(Ap 9) 4gv A= - =2H Peg/Pego' (11)tz tz t

where the Poisson equation

V2 (a0) = -4rGA g (12)

has been used and where

Hg = ( t eg (13)\ 8nGpego

is a scale height for the gas in the absence of stars. Equation (11)

may be solved by the method of separation of variables, in which case

e is written as

e(r,z) = X(z) Y(r). (14)

After some simplification, equation (11) becomes

(Y"(r) + Y(r)) = k2 (X"(z)+X(z) 1/2H2 P (z)/P (15)= g eg ego (15)

Y(r) X(z)

The separation constant k2 arises because the left-hand side of

equation (15) is a function only of r while the right-hand side is a

function only of z.

The differential equation for Y(r)

Y"(r) + r Y'(r) + k2y(r) = 0 (16)

8

has the solution

Y(r) = JO(kr), (17)

where Jo is the zero order Bessel function. The other linearly indepen-

dent solution to equation (16) - - as r - 0 and is therefore not

considered further. The differential equation for X(z) = Xk(z) is

Xz)+i g()pg-) k) (18)Xz) +(2 g (Z)/Pego-k2) Xk(Z)

=

0, (18)

with the boundary conditions

XL(O) = 0

lim Xk(z) = 0.

In addition, Xk(O) is normalized to unity.

because Apg(r,z) is an even function of z.

constraint that Apg(r,z)/Peg(z) - 0 as Izl

The general solution to equation (11)

integral over k of the product Xk(z)JO(kr)

amplitudes A(k)

(19)

(20)

Equation (19) arises

Equation (20) expresses the

can be expressed as an

with the appropriate expansion

e(r,z) = r A(k)Xk(z)Jo(kr)dk,0

(21)

where Jo(kr) is the solution to equation (16) and Xk(z) is the solution

to equation (18) subject to the boundary conditions imposed by equations

(19) and (20). Equations (10) and (21) can be combined to give an

expression for the perturbed gas potential:

APg(r,z) = <v2z> ~ A(k)Xk(z)J(kr)dk - z> C. (22)

9

The solution of the Poisson equation for the perturbed gas potential

is just the sum of the particular solution of the inhomogeneous equation

and the general solution of the homogeneous equation. Equation (22)

with C=O corresponds to the former. The general solution of the homo-

geneous equation is equivalent to the general solution of the Laplace

equation

V2&P 0 (23)

where the subscript h denotes homogeneous.

To solve equation (23) we expand APgh in terms of Jo(kr) with

expansion amplitudes Sk(Z):

Apgh(rz) = <Vz> ro sk(z) Jo(kr)dk. (24)

Expressed in terms of cylindrical coordinates, equation (23) assumes

the form

Ir r (r r aPgh )+ gh = 0. (25)

Equation (24) is substituted into equation (25), and after some

differentiation and algebraic manipulation we find that

= r , ,k J(kr ) Jo k(z) Jo(kr) + kSk(z) r + kJ0(kr dk = O. (26)

We can make use of the well-known relations between the zeroth and

first order Bessel functions Jo and J1 and their derivatives

Jo(kr) = -Jl(kr) (27)

Jo(kr) = -Jo(kr) + Ir Jl(kr) (28)

to simplify equation (26):

10

o = (s(k(z) Jo(kr) - k2sk(z) Jo(kr))dk. (29)

Since the Jo functions are orthogonal, it follows that

sk(z) - k2 sk(z) = 0, (30)

which has the solution

sk(Z) = Wlkek

jz

j + w2ke (31)

The coefficients w2k must vanish if the boundary condition lim APgh = 0

is to be satisfied.

We have now obtained the functional form of Sk(Z) in terms of which

APgh has been expanded. Substitution of equation (31) into equation (24)

easily gives

2 CO , -kJZ (kr)dk. (32)&Pgh(rz) = <tz> wlk J(kr)dk. (32)

Note that Apgh(r,z) expressed here satisfies the boundary condition that

the perturbed gas potential be symmetric with respect to the symmetry

plane z = 0. Adding equations (22) (with C = 0) and (32) then gives the

general solution to the Poisson equation satisfying, of course, the gas

hydrostatic equilibrium equation:

2 co -kiziAPg (r,z) = <vtz> w (A(k)Xk(z) + Wlk

ee ) Jo(kr)dk. (33)

For the sake of completeness we may easily derive from equations

(8) and (21) an expression for the perturbed gas density satisfying

the Poisson and gas hydrostatic equilibrium conditions:

Apg (r,z) = Peg(Z) S0 A(k) Xk(z) Jo(kr)dk. (34)

11

It is appropriate here to consider more closely the nature of

equations (18), (19), and (20) as regards the nature of the marginally

unstable state. Equations (18) - (20) define what is known as the Sturm-

Liouville problem from the theory of differential equations. This is

the problem of determining (a) the relation between the separation constant

k and the function Xk(z) and (b) the influence on k of the boundary conditions

imposed on Xk(z). Stated more simply, we are searching for those values

of k that lead to Xk(z) satisfying both equation (18) and the boundary

conditions imposed by equations (19) and (20). It is worthwhile to note

the mathematical similarity of our problem and the quantum mechanical

problem of determining the allowed energies E of a particle bound by a

one-dimensional potential well V(z). The particle wave function Y(z),

analogous to Xk(z), satisfies the one-dimensional time-independent

Schrodinger equation similar to equation (18), and is subject to the

square integrability condition that r 'Y{2(z)dz be finite, similar to2

equation (20). To carry the comparison one step further, k corresponds

to the particle energy E and 1/2H P ()/p corresponds to the formPeg(z)/Peg° corresponds to the form

of the potential well V(z). Only certain discrete values of k will

yield Xk(z) that obey the boundary conditions. Since Apg(r,z)/p eg(z) =

So A(k) Xk(z) Jo(kr)dk, the statement Xk(z) - 0 as Izl C I is equivalent

to the statement Apg(r,z)/pg(z) - 0 as Izil - . It is the latter

condition then that makes this an eigeenvalue problem.

The stellar component will make its presence felt only as it

effects the depth and or width of the potential well 1/2Hg Peg (z)/Pegog eg ego'

12

or equivalently, as it effects the distribution of gas Peg(Z) above the

galactic plane. It is at this point then that the present stability

analysis differs from the earlier work of Ledoux (1951). Ledoux considered

an isothermal layer of gas (no stars), so that the density distribution

obeys the well-known relation Peg(z)/Pego = sech (z/H). When a stellar

component is included, Peg(z)/Pego can be found by solving simultaneously

the gas and stellar hydrostatic equilibrium equations and the Poisson

2 1/2equation and by choosing appropriate values for Pego' P 0o, <vtz> , and

<v2 >1/2 This has been done in Paper I, from which we recall equation (11):

n Peg(z)/pego - 4G eg(z) + P*o(Peg(Z)/g (35)dz <Vt ego

The observed values Pego = 1H atom/cm3 = 0.025 Me/pc3 (Weaver 1970),

p*o= 0.064 M/pc3 (Luyten 1968), and <v z> = 18 km/sec (Woolley 1958)

2 1/2are chosen. <vtz> is allowed to vary between 0 and 20 km/sec.

At this point we digress slightly to discuss the numerical approach

employed in computing the discrete value or values of the separation

constant k. It can be shown (see Appendix, equation (A16)) that the

solution to equation (18) in the limit z - + - is just

(+) -kz (+) +kzXk(Z) = Qk e + Bk e (36)

We wish to find the restriction on k and Xk(z) if Ok(+)is to vanish.

If Pk = 0, it is clear that

Xk(z) + kXk(z) = 0. (37)

To derive the eigenvalue k from the restriction dictated by equation (37),

a sufficiently large z is considered, denoted by Zc, for which equation

13

(36) holds with good accuracy. Employing numerical methods, equations

(18) and (35) are solved simultaneously for Xk(z), allowing us to tabulate

Xk(Zc) and Xk(zc). The quantity P = Xk(zc) + kXk(Zc) is then calculated

at each of four equally spaced points covering the range of k. The

eigenvalue k lies in the interval over which P changes sign. This

interval is divided into three subintervals and the process repeated

until the desired accuracy is attained.

To determine how the eigenvalue k relates to the linear size of the

neutral state, we recall equation (17) which expresses the solution to

the second order differential equation for Y(r):

Y(r) = JO(kr). (17)

We note that k must have units of reciprocal length, and that for a

given value of k the first zero in Y(r) occurs when

rl = Ql/kl, (38)

where al is the first zero of the zero order Bessel function Jo and k1

is the smallest eigenvalue found by the numerical methods enumerated

above. More generally, if more than one eigenvalue k is found, we

have that

rn = an/kn, (39)

where an is the nth zero of Jo. Focusing on equation (38) we observe

that kl determines the radius of the 'perturbation', or more accurately

the radius r1 of the marginally unstable state in the plane of symmetry

z=O, since

14

e(r1 = cl/kl, z=O) = Xk(0) Jo ('l) = APg(al/kl, 0)/Peg(0) = 0. (40)

When equations (18) and (35) are solved simultaneously subject to

the boundary conditions imposed by equations (19) and (20) and subject

to the values chosen for Pego' P . < v 2 >1/2, and <v2z>l/2 only one

2 1/2eigenvalue k is found. Since <vtz> is allowed to vary between 0 and

2 1/220 km/sec, k1 is determined as a function of <vtz> / 1 may be found as

2 1/2a function of <vtz> from the relation r1 = al/kl. The results are

2 1/2plotted in Figure 1. We see that rl is directly proportional to <vtz

2 1/2and increases from 1 to 2 kpc as <vtz> varies between about 8 and

16 km/sec.

2 1/2 2 >1/2It is useful to inquire how p ego' POX <VZ> and <V tz> effect

kl and thus r1. Recall that the equivalent potential well has the form

2 2V(z) = 1/2Hg Peg(z)/Pego.' The central depth is just V(0) = 12Hg since

Peg(0)/Pego= 1. Since Hg is a function only of Pego and <vtz>, Po and

<v2 > have no effect on the central depth. Increasing the depth of the*z

2potential well by decreasing Hg (and thus increasing Pego or decreasing

2 2<vtz>) will increase k1 and decrease rl. p*O and <vsz> will, however,

effect the width of the potential well, in the sense that increasing

p o or decreasing <v2 > decreases the width which causes k1 to decrease *0 .z

and r1 to increase. In summary, the depth is determined only by Pego

2and <vtz>, while the width is determined primarily by p*o, and to a

2 2lesser extent by Pego <v >, and <Vtz> . The inclusion of a stellarego, *Z tz

15

component will increase the radius rl of the marginally unstable state

above its value when p = 0.

It is instructive to compare rl calculated with Pego = 0.025 3/pc3

P*o = 0.064 MV/pc3 , and <v2 >1/2 = 18 km/sec with the Jeans' radius

(XJ/2) and the Ledoux radius (X L/2), each calculated with Pego = 0.025

2 1/2OD/pc 3 . <vtz> is a free parameter, allowed to vary bet een 1 and 20

km/sec. The results are presented in Table 1. It is particularly useful

to compare rl (Kellman) to rl (Ledoux), since both assume plane-parallel

gas distributions in the equilibrium state. The presence of a rigid

stellar component with p*, = 0.064 MI/pc3 and <v2 >1/2 = 18 km/sec

increases the radius of the marginally unstable state, though only slightly.

Isodensity contours of the marginally unstable or neutral state

may be derived by combining equations (1) and (34), with the result that

pg(r,z) Peg(Z)-pg -e= g) (1 + A(k) Xk(z) JO(kr)). (41)ego ego

The integral over k in equation (34) reduces to just one term since

only the neutral state is being considered and since only one eigenvalue

was found. At the center of the perturbation r = z = 0, Xk(z) = 1,

Jo(kr) = 1, and Peg(Z)/Pego = 1, and equation (41) becomes

p (O,O)9Pg O = 1 + A(k). (42)

Pego

21-cm observations of large gas structures in spiral arms (McGee and

Milton 1964) indicate that 1.0 ' Pg(0,0)/Pego ! 2.0, and therefore

16

0 S A(k) ! 1.0. Making use of equation (35) for Peg(Z)/P,go, equation

(18) for Xk(z), and numerical tables for Jo(kr), equation (41) is used

to construct an (r,z) grid of values for Pg(r,z)/pego from which

contours of equal Pg(r,z)/lpego may be derived. Figure 2 displays three

isodensity contours pg(rz)/Pego = 1.3, 0.7, and 0.3 calculated with

A(k) = 0.5. The neutral state is clearly quite flattened, and is

reminiscent of contours presented by McGee and Milton (1964) and

Westerhout (1956), the latter referring to cross sections of spiral arms.

The mass of the marginally unstable state may be expressed as an

integral of the gas density over a cylinder perpendicular to the galactic

plane, of infinite height and of radius rl = l/kl:

+rlMg = 2r-r c 1 Pg(r,z)r dr dz. (43)

Equation (43) can be evaluated numerically with the aid of equation (41).

Choosing <vt1/2= 10.0 km/sec (see Paper I) and values quoted previously

for p ego, p , and<v >1/2, we find that Mg 5xlO6 M.

Equations (18) and (35) may be written in dimensionless form

d~2 Xk = _ ( 2 Ck )Xk

d 2 2

where the dimensionless quantities a, and are defined

where the dimensionless quantities Ckfoll, ow, s:, and are defined

as follows:

17

Ck = kHg (46)

= Peg(Z)/Pego (47)

= z/Hg (48)

= P*o/Pego (49)

6 = <v >/<v2>. (50)

tz *z

The radius r1 of the neutral state in the plane of symmetry can be

2 1/2written as r1 = 405Hg but since Hg = (<v >/8Gpego) ,r depends

Con the ratio (< >/ego

2 1/2on the ratio (<v t>/p ego) precisely the functional dependence found

2 1/2by Jeans and Ledoux. If <vtz> is independent of distance from the

galactic center beyond the solar distance Ro (see Paper I), r1 would be

expected to increase with increasing distance from the galactic center

since Pego decreases with increasing R. Westerhout (1956) finds that

(corrected to the new galactic distance scale) pego (R = 11 kpc)/p ego(R = 15 kpc)

P4.7 from which it follows that rl(R = 15 kpc)/rl(R = 11 kpc)

t 2.2 due to a change in p ego alone. In conclusion, since rl = 2.405 Hg,Ck

rl depends principally on <v2 > 1 /2

and p through H , to a smaller

extent on p through C, and to an even smaller extent on <v2 >1/2extenthrough Ck and to an even smaller extent on <V

through Ck.

18

III. COMPARISON WITH THE OBSERVATIONS

It is interesting that a 21-cm southern survey by McGee and Milton

(1964) revealed that the principal elements of the gaseous component of

spiral arms in our galaxy are large flattened structures with dimensions

107 Me and 1-2 kpc, strung out along the length of the arms much like

beads on a string. More recently, Kerr (1968) and Westerhout (1971)

have pointed to the existence of large gas condensations forming the

major components of spiral arms. The size and mass of the individual

gas structures observed by McGee and Milton are close to the values

we have calculated for the marginally unstable state. Further, their

linear size is observed to increase markedly with increasing distance

from the galactic center beyond Ro, consistent with our findingsabove,

while their mass is observed to remain essentially constant at about

107 MD. In addition, they occur predominantly in the outer arms beyond

Ro (McGee 1964). It is also of interest that McGee and Milton (1966)

have observed similar gas structures in the Large Magellanic Cloud with

a mean linear size and mass of about 600 pc and 4x106 MO, respectively.

2 1/2The smaller size may indicate that Pego is larger, <vtz> is smaller,

and or p*o is smaller than corresponding values in our galaxy, if the

theory is to be believed.

IV. DISCUSSION

Recent work by Lin (1970) has refocused attention on the importance

of classic Jeans' type stability analyses of the gaseous component of

galaxies. It is Lin's opinion that density waves (primarily in the

19

stellar component) are initiated or excited by the gravitational

instability of the gas in the outer regions of galaxies, where the

star density no longer overwhelmingly dominates the gas density and where

the gas turbulence, which tends to inhibit instability, may be small

because of reduced energy inputs. Excited in the outer regions, the

density waves travel inward, subsequently giving rise to two-armed

spiral patterns. Once formed, the large gas condensations are stretched

by differential galactic rotation into trailing structures, and because

they lie near or outside the corotation distance, form material arms.

The pattern speed Qp over the whole galactic disk is determined then, if

Lin is correct, by the angular velocity of these outer gas condensations

around the galactic center. In support of Lin's ideas, Shu, Stachnick,

and Yost (1971) obtained a good fit to the spiral structure observed

in M33, M51, and M81 by assuming that Qp is equal to the angular rotation

speed of the outermost HII regions.

We mentioned above that the large gas condensations observed by

McGee and Milton (1964) are preferentially located at distances beyond

the solar distance Ro. Not only does this fact agree nicely with Lin's

ideas, but it follows directly from results obtained in Paper I of this

series. There we found evidence that the rms z turbulent gas velocity

dispersion <v2 1/2 (or more correctly Q = (<vz> + Bo2 /8p ego+ Pcrpego) /2)

increases with decreasing distance from the galactic center in the range

4 kpc < R $ 10 kpc. It follows then that gravitational instability of

the gas disk is inhibited when R < Ro .

20

ACKNOWLEDGMENTS

The author is grateful to Professor J. P. Ostriker and Professor

G. B. Field for their advice and encouragement, and to Dr. L. A. Fisk

for a critical reading of the manuscript. This work was supported in

part by a National Science Foundation Graduate Fellowship while the

author was a student at the University of California at Berkeley, and

by NASA Grant No. NGL 21-002-033.

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Goldreich, P., and Lynden-Bell, D. 1965, M.N.R.A.S., 130, 97.

Goon, G. 1966, private communication.

Jeans, J. H. 1928, Astronomy and Cosmology, p. 337.

Kellman, S. A. 1972a, in preparation.

Kellman, S. A. 1972b, in preparation.

Kellman, S. A. 1972c, in preparation.

Kerr, F. J. 1968, Nebulae and Interstellar Matter, ed. B. M. Middlehurst

and L. H. Aller (Chicago: The University of Chicago Press), p. 575.

Ledoux, P. 1951, Ann. d'Ap., 14, 438.

Lin, C. C. 1970, The Spiral Structure of Our Galaxy, ed. W. Becker and

G. Contopoulos (Dordrecht: D. Reidel Publishing Co.), p. 373.

Lin, C. C., and Shu, F. H. 1964, Ap. J., 140, 646.

Lin, Co C. 1966, Proc. Nat. Acad. Sci., 55, 229.

Lin, C. C., Yuan, C., and Shu, F. H. 1969, Ap. J., 155, 721.

Luyten, W. J. 1968, MoN.R.A.S., 139, 221.

McGee, R. X. 1964, The Galaxy and the Magellanic Clouds, ed. F. J. Kerr

and A. W. Rodgers (Canberra: Australian Academy of Science), p. 126.

21

McGee, R. X., and Milton, J. A. 1964, Aust. J. Phys., 17, 128.

McGee, R. X. 1966, Aust. J. Phys., 19, 343.

Shu, F. H., Stachnik, R. V., and Yost, J. C. 1971, Ap. J., 166, 465.

Weaver, H. F. 1970, private communication.

Westerhout, G. 1956, B. A. N., 13, 201.

Westerhout, G. 1971, private communication.

Woolley, R. vod. R. 1958, M.N.R.A.S., 118, 45.

22

APPENDIX

ALTERNATE CALCULATION OF Lpg(r,z)

As a check on the correctness of equation (22), Acpg(r,z) may be

derived by a different approach, beginning directly with the solution

to the Poisson equation for the perturbed gas potential. Expressed in

cylindrical coordinates, we have that

ACpg(r °, 0 z) = 2G o o 1 Apg(r, 0, z)rdrd0dz. (Al)

(rO, 00, zo) and (r, 0, z) are the coordinates of any two points and

R is the distance between them. It can be shown that

= E Cmcos[m(0-0O)] o Jm(kr) Jm(kro) e dk, (A2)R m=0 m o

and noting that

f cos[m(0-0o)]d0 = 2T6 mo, (A3)

where 6 is the Kronecker delta symbol, it follows that

AcPg(ro, zo) = 2rG rdr 7_- dz Apg(r,z) J(kr)J(kr)J(kro)e Idk (A4)

The perturbed gas density Apg(r,z) may be expanded in terms of the zero

order Bessel function Jo(k'r) and the appropriate expansion amplitudes

Tk (z)

Apg(r,z) = So Tk'(z) Jo(k'r) dk', (A5)

and if we note that

r Jo(kr) Jo(k'r)dr = k 6(k'-k), (A6)

it follows that

Acpg(ro,zo) = 2rrG Lodz 1 Tk(z)e klz-zoIJo(kr)dk. (A7)

23

Note that two expressions have been derived for Apg(r,z), represented

by equations (34) and (A5). If they are equated we find that

Tk(z) = A(k)Peg(Z)Xk(Z), (A8)

and equation (A7) becomes

ACpg(ro, zo) = 2rrGJ' dk A(k)Jo(kro) Pg z)e- Zoldz. (A9)

Equation (18) can be recalled to supply an expression for the quantity

Peg(z)Xk(z) appearing in equation (A9):

Peg(Z)Xk(z) = 2 PegoHg (k Xk(z)-Xk(z)). (A10)

It is this condition that, when substituted into equation (A9), constrains

ACpg(ro, Zo) to satisfy the gas hydrostatic equilibrium equation:

A~g(ro, Zo) G2 Ao (k)Jo(kro)1 °

2 i -k Iz-zol2 g(r o) = TTGHg PepgoJ0 dk k '- (k Xk(z)-Xk(z))e dz. (All)

To proceed further it becomes necessary to evaluate the integral

over z in equation (All). This can be accomplished by splitting the

range into two parts:

Jc (k Xk(z)-Xk(z))e -zoldz =- Xk(z)e+k(z-Zo)dz

+ k 2 J X()e k(z-zdz - J Xk()e-k(z -zO)dz + k2 o X -k(z-)e dz. (A12)J -00 kzoe dz - zo k, zde Is z0

The first and third terms on the right hand side of equation (A12) are

integrated twice by parts and added to the second and fourth terms, and

after some simplification we find that

coc (k2Xk( z )-Xk(z ))e

- z° l

d z = -Xk( z )e+ k ( z - z o) zo

(A13)

+k(z-z) 0 e-k(z-zo) co , -k(z-zo)i o+ kXk(Z)e k -Xk(Z)e kk(z)e

24

To evaluate these terms in the limit z - + o we recall the differential

equation for Xk(z):

Xk(z) + Peg(Z)/Pego k2) Xk(Z) = 0. (A14)

In the limit z -_+ _, equation (A14) becomes

Xk(z) = k2 Xk(z), (A15)

since £im 1z -+co 2H2 Peg(Z)/Pego = 0. The solution to equation (A15)

g

can easily be written:

(+) -kz (+) +kzz > 0 : Xk(z) = Gk e + B k e (A16)

z < 0: Xk(Z) = k(-)e-kZ+ k(-)e+

k z (A17)

Referring back to.equation (34) for the perturbed gas density

Apg(r,z)/Peg(z) = Jo A(k)Xk(z)Jo(kr)dk, (A18)

it is clear that unless Yim Xk(z)=0,the quantity Apg(r,z)/peg(z) will notz + Xk(z)=O te

0 as z _ + o. This requires that ak

(

= = 0. Symmetry of

Apg(r,z)/Peg(z) about the plane z - 0 results in the further restriction

that ak( ) = Ok(-) Equations (A16) and (A17) therefore become

-kzz > 0: Xk(z) = ye (A19)

z < 0: Xk(z) = ye+kz, (A20)

where y = k(+ ) = (- It follows that

25

z > 0: Xk(z) = -yke (A21)

+kzz < 0: Xk(z) = yke o (A22)

We can now evaluate each term in equation (A13) at the infinite limit.

For example, cofisider the 'first term:

eim (-X(z)ek(ZZo)) = yim (-ke-kZoe+2kz)= ° (A23)Z - -CO Z -+ -OO

Similarly, the other three terms evaluated at the infinite limit also

vanish. We are now able to evaluate the integral over z appearing in

equation (All):

(k2Xk(z) klz-ZoiJ -o(k2Xk) - Xk(z))e kdz = -Xk(zo)+kXk(Zo)+Xk(o)+kXk(o) = 2 kXk(Zo).

(A24)

Equation (All) therefore simplifies to

AbPg2 (rz) = 8 2GH Pego o A(k)Xk(z)JO(kr)dk, (A25)

where the arbitrary coordinates (ro, zo) have been replaced by (r,z).

This is, however, just the particular solution to the Poisson equation

(and consistent with the gas hydrostatic equilibrium equation), for which

we have derived another expression in the way of equation (22)

(with C = 0):

Acpgl(r,z) = <vtz> 0 A(k) Xk(z)JO(kr)dk. (A26)

2 2Recalling that H <vtz>/8 Pego) it is clear thatg tz/8rpego,

AcPgl(r,z) = Apg2 (r,z). (A27)

The purpose of the above derivation was to obtain an independent

check on Acpgl(r,z) derived in Section II, and the equality expressed by

equation (A27) gives us confidence that the expression obtained for

ACPg(r,z) is correct.

26

TABLE 1

COMPARISON BETWEEN THE JEANS', LEDOUX, AND2 1/2

KELLMAN RADIUS AS FUNCTIONS OF <vtz>

V2 tz rl(Jeans) rl(Ledoux) rl(Kellman)

(km/sec) (kpc) (kpc) (kpc)

1.0

2.5

5.0

7.5

10.0

15.0

20.0

0.085

0.213

0.425

0.638

0.850

1.275

1.700

0.120

0.301

0.601

0.902

1.202

1.803

2.404

0.127

0.318

0.635

0.953

1.270

1.905

2.540

27

FIGURE CAPTIONS

1. The dimension rl of the marginally unstable state in the symmetry

plane (z = 0) as a function of the rms z turbulent gas velocity

dispersion.

2. Isodensity contours pg(r,z)/pego = 1.3, 0.7, and 0.3 of the

marginally unstable state, calculated with A(k) = 0.5.

3

2

r, (kpc)

005 10 15

<V2 >/2 (km/sec)Figure 1

J

20

z (pc)

r(pc)

Figure 2